Nonlinear Dynamical Systems Fifth Class › pessoas › ... › Aulas › Aula05-english.pdf · Gradient semigroups Dynamically gradient semigroups Nonlinear Dynamical Systems Fifth

Gradient semigroupsDynamically gradient semigroups

Nonlinear Dynamical SystemsFifth Class

Alexandre Nolasco de Carvalho

August 31, 2017

Alexandre N. Carvalho - USP/Sao Carlos Second Semestre of 2017


Gradient semigroups

In this section we consider the gradient semigroups. This class ofsemigroups appears naturally in several applications and itscharacteristics allow us to describe with precision the structure ofits attractors.



DefinitionA semigroup T (t) : t ≥ 0 with an invariant set Ξ is gradientrelatively to Ξ if there is a continuous function V : X → R suchthat

(i) R+ 3 t 7→ V (T (t)x) ∈ R is decreasing for each x ∈ X ;

(ii) If x is such that V (T (t)x) = V (x) for all t ∈ R+, then x ∈ Ξ.

(iii) V is constant in each connected component of Ξ.

A function V : X → R with these properties is called a Lyapunovfunction for T (t) : t > 0.



Gradient Vector Fields

The notion of gradient semigroups is associated to those vectorfields which are gradient of a potential; that is, if φ : Rn → R is aC 1 potential with Rn 3 u 7→ ∇φ(u) ∈ Rn locally Lipschitzcontinous, the differential equation

u = −∇φ(u), t > 0,

u(0) = u0 ∈ Rn,(1)

is called a gradient system.



The potential φ has the property that, if [0, τu0) 3 t 7→ u(t, u0) isa solution of (1) then [0, τu0) 3 t 7→ φ(u(t)) ∈ R is decreasing andconsequently, does not blow up as t → τ−u0

.

If φ(x)→∞ as ‖x‖Rn →∞ we must have that τu0 = +∞ for allu0 ∈ Rn, and we can define the semigroup T (t) : t > 0 byT (t)u0 = u(t, u0), t ≥ 0 and u0 ∈ Rn.

It is easy to see that φ(u(·)) is constant if and only if u(·) isconstant and ∇φ(u) = 0 showing that φ : Rn → R is a Lyapunovfunction for T (t) : t ≥ 0 relative to the set Ξ of equilibria for(1); that is, Ξ = u ∈ Rn : ∇φ(u) = 0.



A Heat Equation

Let Ω be a bounded smooth domain in RN , N > 1. Consider theparabolic problem

ut = ∆u + f (u), x ∈ Ω, t > 0,

u(x) = 0, x ∈ ∂Ω, t > 0,

u(0) = u0 ∈ H10 (Ω).

(2)



Assume that f : R→ R is a globally Lipschitz function withbounded derivatives up to second order. Then, there is asemigroup T (t) : t > 0 associated to (2). The functionalV : H1(Ω)→ R given by

V (u) =1

2

∫Ω|∇u|2 +

∫ΩF (u), (3)

where F (u)(x) = −∫ u(x)

0f (s)ds, is a Lyapunov function for

T (t) : t > 0.



It is easy to see that V (u(·)) is constant if and only if u(·) isconstant and u ∈ H2(Ω) ∩ H1

0 (Ω) is a solution of

0 = ∆u + f (u), x ∈ Ω,

u(x) = 0, x ∈ ∂Ω.(4)

showing that V : H10 (Ω)→ R is a Lyapunov function for

T (t) : t ≥ 0 relative to the set Ξ of equilibria for (2); that is,Ξ = u ∈ H2(Ω) ∩ H1

0 (Ω) : u satisfies (4).



A Damped Wave Equation

Let Ω be a bounded smooth domain in RN , N > 1. For β > 0,consider the damped hyperbolic problem

utt + 2βut = ∆u + f (u), x ∈ Ω, t > 0,

u(x) = 0, x ∈ ∂Ω, t > 0,

u(0) = u0 ∈ H10 (Ω), ut(0) = v0 ∈ L2(Ω).

(5)



This boundary-initial value problem can be written as an abstractordinary differential equation in H1(Ω)× L2(Ω) in the followingway. If A : D(A) ⊂ L2(Ω)→ L2(Ω) is defined by Au = ∆u for allu ∈ D(A) = H2(Ω) ∩ H1

0 (Ω) andΛ : D(Λ) ⊂ H1(Ω)× L2(Ω)→ H1(Ω)× L2(Ω) is defined by

Λ

[uv

]=

[0 I∆ −β

] [uv

]=

[v

∆u − βv

]we may rewrite (5) as

d

dt

[uv

]= Λ

[uv

]+

[0

f (u)

], t > 0,[

uv

](0) =

[u0

v0

]∈ H1

0 (Ω)× L2(Ω).

(6)



Assume that f : R→ R is a globally Lipschitz function withbounded derivatives up to second order. Then, there is asemigroup T (t) : t > 0, in H1(Ω)× L2(Ω) associated to (6).The functional V : H1(Ω)× L2(Ω)→ R given by

V

([uv

])=

1

2

∫Ω|∇u|2 + β

∫Ωv2 +

∫ΩF (u), (7)

where F (u)(x) = −∫ u(x)

0f (s)ds, is a Lyapunov function for

T (t) : t > 0.



It is easy to see that V

([uv

](·))

is constant if and only if

[uv

](·)

is constant, v = 0 and u ∈ H2(Ω) ∩ H10 (Ω) satisfies (4) showing

that V : H10 (Ω)× L2(Ω)→ R is a Lyapunov function for

T (t) : t ≥ 0 relative to the set Ξ of equilibria for (6); that is,

Ξ =

[u0

]: u ∈ H2(Ω) ∩ H1

0 (Ω) satisfies (4)

.



For gradient semigroups we have the following characterizationresult.

LemmaLet T (t) : t > 0 be a gradient semigroup relative to an invariantset Ξ. Then, ω(x) ⊂ Ξ for each x ∈ X and, if there is a globalsolution φ : T→ X through x , then αφ(x) ⊂ Ξ. Furthermore, if Ξis disjoint union of closed invariant sets Ξ1, · · · ,Ξn, thenω(x) ⊂ Ξi for some 1 ≤ i ≤ n and if here is a global solutionφ : T→ X through x , then αφ(x) ⊂ Ξj for some 1 ≤ j ≤ n.



Proof: If y ∈ ω(x), there is a sequence tnn∈N, with tnn→∞−→ ∞,

such that y = limn→∞ T (tn)x . From the continuity of V we havethat V (y) = limn→∞ V (T (tn)x). Since T+ 3 t 7→ V (T (t)x) ∈ Ris non-increasing we have that V (y) = limt→∞ V (T (t)x). Fromthis and from the continuity of V

V (y) = limn→∞

V (T (t + tn)x) = limn→∞

V (T (t)T (tn)x) = V (T (t)y),

for each t ∈ T+. From the property (ii) in Definition 1 we havethat ω(x) ∈ Ξ.



Suppose that there exists a global solution φ : T→ X through x .With the same reasoning as above, if y ∈ αφ(x), we have thatV (y) = limt→∞ V (φ(−t)).

From this and from the continuity of V we have thatV (T (t)y) = V (y), t ∈ T+, and therefore y ∈ Ξ.

The last statement of the lemma follows from the fact that ω(x)and αφ(x) are connected in the case T = R and as before in thediscrete case (exercise).



TheoremIf T (t) : t > 0 is an is eventually bounded and asymptoticallycompact semigroup which is gradient relatively to a boundedinvariant set Ξ, then T (t) : t > 0 has a global attractorA = W u(Ξ), where

W u(Ξ) := y ∈ X : there is a global solution φ : T→ X

with φ(0) = y such that φ(t)t→−∞−→ Ξ

is called the unstable set of Ξ. If Ξ =⋃n

i=1 Ξi whereΞ = Ξ1, · · · ,Ξn is a disjoint collection of closed invariant sets,then A = ∪ni=1W

u(Ξi ). Finally, if there is a bounded connectedset B that contains A, then A is connected.



Proof: Since T (t) : t > 0 is eventually bounded andasymptotically compact, for each x ∈ X , ω(x) is non-empty,compact, invariant and attracts x .

From the fact that T (t) : t > 0 is gradient, it follows thatω(x) ⊂ Ξ and, since Ξ is bounded, we have that T (t) : t > 0 ispoint dissipative.

From a previous theorem, T (t) : t > 0 has a global attractor.



If x ∈ A, from a previous lemma there is a global solutionφ : T→ X through x . Since φ(T) ⊂ A is relatively compact,αφ(x) 6= ∅ and from Lemma 2, αφ(x) ⊂ Ξ.

This shows that A ⊂W u(Ξ). If x ∈W u(Ξ), there is a global

solution φ : T→ X through x and φ(t)t→±∞−→ Ξ ⊂ A.

From the fact that φ(T) is bounded and invariant we conclude thatφ(T) ⊂ A and, consequently, x ∈ A. This shows A ⊃W u(Ξ) andcompletes the proof that A = W u(Ξ).



If Ξ =⋃n

i=1 Ξi with Ξ1, · · · ,Ξn a disjoint collection of closedinvariant sets, the equality A = ∪ni=1W

u(Ξi ) follows immediatelyfrom Lemma 2 and, if A is contained in a bounded connectedsubset of X , it follows from the invariance of ω(B), from the factthat B ⊃ A and from a previous lemma that ω(B) = A isconnected.



The following lemma is an immediate consequence of thecontinuity of semigroups and it is an important tool for the proofof the results that follow.

It ensures that, given an invariant set Ξ for the semigroupT (t) : t > 0 and y close to Ξ, T (s)y : 0 6 s 6 t remainsclose to Ξ for large values of t.



LemmaLet T (t) : t > 0 be a semigroup and Ξ be a compact invariantset for T (t) : t > 0. Given t > 0 and ε > 0, there exists δ > 0such that T (s)y : 0 6 s 6 t, y ∈ Oδ(Ξ) ⊂ Oε(Ξ).



Proof: Suppose that there are t > 0 and ε>0 such that, foreachk∈N∗ there are xk ∈O1

k(Ξ) and sk∈ [0, t ] with d(T (sk)xk ,Ξ)> ε.

We may assume that skk→∞−→ s0 for some s0 ∈ [0, t ] and that

xkk→∞−→ y ∈ Ξ. Since T+×X 3 (t, x) 7→ T (t)x ∈ X is continuous,

we have that 0 = d(T (s0)y ,Ξ) > ε which is an absurd.



LemmaLet T (t) : t > 0 be a gradient semigroup relative to an invariantset Ξ. Suppose that T (t) : t > 0 has a global attractor A, thatΞ =

⋃ni=1 Ξi with Ξ = Ξ1, · · · ,Ξn being a disjoint collection of

closed invariant sets, n ∈ N∗, and that the associated Lyapunovfunction is constant on each Ξi , 1 ≤ i ≤ n. LetV (Ξ) = n1, · · · , np with ni < ni+1, 1 6 i 6 p − 1.



If 1 6 j 6 p − 1 and nj 6 r < nj+1, then Xr = z ∈ X : V (z) 6 ris positively invariant under the action of T (t) : t > 0 andTr (t) : t > 0, the restriction of T (t) : t > 0 to Xr , has aglobal attractor A(j) given by

A(j) = ∪W u(Ξ`) : V (Ξ`) 6 nj.

In particular, V (z) 6 nj for z ∈ A(j), n1 = minV (x) : x ∈ X andA(1) = ∪Ξ ∈ Ξ : V (Ξ) = n1 consists of all asymptotically stableinvariant sets; that is, for each Ξ ∈ Ξ with Ξ ⊂ A(1) there is anε > 0 such that T (t)x

t→∞−→ Ξ whenever x ∈ Oε(Ξ).



Proof: It is clear from the definition of a Lyapunov function thatXr is positively invariant under the action of T (t) : t > 0.

To show the existence of a global attractor for Tr (t) : t > 0 wenote that the required properties for the existence of a globalattractor are inherited from those of T (t) : t > 0; namely, orbitsof bounded subsets of Xr are bounded, Tr (t) : t > 0 is pointdissipative and asymptotically compact.

Hence, Tr (t) : t > 0 has a global attractor A(j). The restrictionVr of V to Xr is a Lyapunov function for Tr (t) : t > 0associated with Ξ` : V (Ξ`) 6 nj and the characterization of A(j)

follows.



Let us now prove the last claim. Let δ0 = 12 mind(Ξ, Ξ) > 0

where the minimum is taken over the sets Ξ, Ξ ∈ Ξ such thatΞ, Ξ ⊂ A(1) and Ξ 6= Ξ.

Assume by contradiction that there exist Ξ 3 Ξ ⊂ A(1), 0 < δ < δ0

and sequences xkk∈N in X , tkk∈N in T+ such that xkk→∞−→ Ξ,

d(T (t)xk , Ξ) < δ for 0 6 t < tk and d(T (tk)xk , Ξ) = δ.

From Lemma 4 we have that tkk→∞−→ ∞ and the asymptotic

compactness of T (t) : t > 0 ensures that T (tk)xkk∈N has aconvergent subsequence, which we denote the same, and let y beits limit.



It is easy to see that n1 is the minimum value of V in X , and fromthis is immediate that

n1 = V (Ξ) = limk→∞

V (xk) > limk→∞

V (T (tk)xk) = V (y) > n1,

and

n1 = V (Ξ) = limk→∞

V (xk)

> limk→∞

V (T (t + tk)xk) = V (T (t)y) > n1, for each t ∈ T+.



Hence V (y) = V (T (t)y) = n1 for all t ∈ T+ and therefore y ∈ Ξ.Since d(y , Ξ) < δ0 we have that y ∈ Ξ. But d(y , Ξ) = δ > 0 andgive us a contradiction.

This proves that for each Ξ ∈ A(1) and 0 < δ < δ0 there is a0 < δ′ < δ such that, for all x ∈ Oδ′(Ξ), we have thatγ+(x) ⊂ Oδ(Ξ) and proves that A(1) consists only of stableinvariant sets.

To conclude, we only need to note that, for each x ∈ X ,T (t)x

n→∞−→ Ξ for some Ξ ∈ Ξ.


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Nonlinear Dynamical Systems Fifth Class › pessoas › ... › Aulas › Aula05-english.pdf · Gradient semigroups Dynamically gradient semigroups Nonlinear Dynamical Systems Fifth